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curve #1568

y² + xy = x³ + 386484616276x + 155761394676098448

a-invariants: [1, 0, 0, 386484616276, 155761394676098448]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/8ℤ
conductor (N): 671144334 ☆ record for ℤ/8ℤ torsion, rank ≥ 3
discriminant (Δ): -14175690207451086775627113800147140608
Faltings height: 5.8129
naive height: 92.6973
primes of bad reduction: 2, 3, 7, 19, 41, 73, 281
regulator: 158.11615123393688000449442454763423084445664
submitted by: Michael Rubinstein
submitted at: 2026-09-25 01:12:19 UTC
last updated: 2026-09-25 01:12:19 UTC

Witness: 3 independent points log in to add more points →

(68816224/49, 638566099732/343)
(786697048/121, 22167912654908/1331)
(-118043432/2401, 43491876631636/117649)

Commentary

Torsion Z/8. From Michael Rubinstein's github repository: https://github.com/mrubinst/elliptic-curves-by-rank

last edited by Michael Rubinstein at 2026-09-25 01:12:19 UTC · history

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